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Thursday, October 26, 2006

Euler's Enchiladas

Leonhard Euler lived from 1707 til 1783. He published such an astonishing amount of mathematics that the St. Petersburg Academy continued publishing his work for more than 30 years after his death. Eventually he went blind, but continued doing enormous calculations in his head. He could recite the entire Aeneid of Virgil. One thing he did was study the multiple zeta values. He proved the two argument (depth 2) version of the result that the value of the Riemann zeta function at the 1-ordinal n was the sum over (depth k , weight n) MZVs such that the first argument was greater than 1. The depth 3 case was proved in 1996.

Euler's MZVs were largely forgotten until recent times, but since their appearance in QFT structures they have arisen in many contexts. Multiple polylogarithms are a natural generalisation. Now we know that the MZVs are algebraic integrals for the cohomology of moduli of punctured spheres.